Deligne cohomology

In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.

For introductory accounts of Deligne cohomology see Brylinski (2008, section 1.5), Esnault & Viehweg (1988), and Gomi (2009, section 2).

Definition

The analytic Deligne complex Z(p)_{D, an} on a complex analytic manifold X is

$0\rightarrow \mathbf {Z} (p)\rightarrow \Omega _{X}^{0}\rightarrow \Omega _{X}^{1}\rightarrow \cdots \rightarrow \Omega _{X}^{p-1}\rightarrow 0\rightarrow \dots$

where Z(p) = (2π i)^pZ. Depending on the context, $\Omega _{X}^{*}$ is either the complex of smooth (i.e., C^∞) differential forms or of holomorphic forms, respectively. The Deligne cohomology H q
D,an (X,Z(p)) is the q-th hypercohomology of the Deligne complex. An alternative definition of this complex is given as the homotopy limit[1] of the diagram

${\begin{matrix}&&\mathbb {Z} \\&&\downarrow \\\Omega _{X}^{\bullet \geq p}&\to &\Omega _{X}^{\bullet }\end{matrix}}$

Properties

Deligne cohomology groups H q
D (X,Z(p)) can be described geometrically, especially in low degrees. For p = 0, it agrees with the q-th singular cohomology group (with Z-coefficients), by definition. For q = 2 and p = 1, it is isomorphic to the group of isomorphism classes of smooth (or holomorphic, depending on the context) principal C^×-bundles over X. For p = q = 2, it is the group of isomorphism classes of C^×-bundles with connection. For q = 3 and p = 2 or 3, descriptions in terms of gerbes are available (Brylinski (2008)). This has been generalized to a description in higher degrees in terms of iterated classifying spaces and connections on them (Gajer (1997)).

Relation with Hodge classes

Recall there is a subgroup ${\text{Hdg}}^{p}(X)\subset H^{p,p}(X)$ of integral cohomology classes in $H^{2p}(X)$ called the group of Hodge classes. There is an exact sequence relating Deligne-cohomology, their intermediate Jacobians, and this group of Hodge classes as a short exact sequence

$0\to J^{2p-1}(X)\to H_{\mathcal {D}}^{2p}(X,\mathbb {Z} (p))\to {\text{Hdg}}^{2p}(X)\to 0$

Applications

Deligne cohomology is used to formulate Beilinson conjectures on special values of L-functions.

Extensions

There is an extension of Deligne-cohomology defined for any symmetric spectrum $E$ [1] where $\pi _{i}(E)\otimes \mathbb {C} =0$ for $i$ odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.

References

Hopkins, Michael J.; Quick, Gereon (March 2015). "Hodge filtered complex bordism". Journal of Topology. 8 (1): 147–183. arXiv:1212.2173. doi:10.1112/jtopol/jtu021. S2CID 16757713.

Deligne-Beilinson cohomology
Geometry of Deligne cohomology
Notes on differential cohomology and gerbes
Twisted smooth Deligne cohomology
Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups
Brylinski, Jean-Luc (2008) [1993], Loop spaces, characteristic classes and geometric quantization, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4731-5, ISBN 978-0-8176-4730-8, MR 2362847
Esnault, Hélène; Viehweg, Eckart (1988), "Deligne-Beĭlinson cohomology" (PDF), Beĭlinson's conjectures on special values of L-functions, Perspect. Math., vol. 4, Boston, MA: Academic Press, pp. 43–91, ISBN 978-0-12-581120-0, MR 0944991
Gajer, Pawel (1997), "Geometry of Deligne cohomology", Inventiones Mathematicae, 127 (1): 155–207, arXiv:alg-geom/9601025, Bibcode:1996InMat.127..155G, doi:10.1007/s002220050118, ISSN 0020-9910, S2CID 18446635
Gomi, Kiyonori (2009), "Projective unitary representations of smooth Deligne cohomology groups", Journal of Geometry and Physics, 59 (9): 1339–1356, arXiv:math/0510187, Bibcode:2009JGP....59.1339G, doi:10.1016/j.geomphys.2009.06.012, ISSN 0393-0440, MR 2541824, S2CID 17437631

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[:0-1] Hopkins, Michael J.; Quick, Gereon (March 2015). "Hodge filtered complex bordism". Journal of Topology. 8 (1): 147–183. arXiv:1212.2173. doi:10.1112/jtopol/jtu021. S2CID 16757713.